91Ó°ÊÓ

A series is an infinite sum of terms, and determining its convergence is crucial in mathematics. Convergence means that as we add more and more terms, the series approaches a specific finite value. For geometric series, this decision hinges on the common ratio. In our example series \( \sum_{k=1}^{\infty} \left( \frac{1}{7} \right)^{k} \), convergence can be checked using the common ratio \( r \).

To determine if a geometric series converges, we must examine the absolute value of the common ratio \( |r| \). If \( |r| \) is less than 1, the series converges. In this exercise, \( |r| = \frac{1}{7} \), which fits the convergence condition \( |r| < 1 \). Therefore, we conclude that this series converges!

This concept is fundamental because convergent series can be assigned a sum, whereas divergent ones grow indefinitely or oscillate without settling to a sum.

In a geometric series, each term after the first is derived by multiplying the preceding term by a fixed number called the common ratio. Understanding the common ratio is key to mastering geometric series.

For the given series \( \sum_{k=1}^{\infty} \left( \frac{1}{7} \right)^{k} \), the common ratio \( r \) is \( \frac{1}{7} \).

Here's why identifying the common ratio is important:

• It helps in determining whether the series converges or diverges.
• It is critical in finding the sum of the series, if it converges.

The common ratio can be any real number, but the criteria \( |r| < 1 \) is what ensures the series converges. If \( |r| \) were 1 or greater, the series would diverge, leaving no finite sum to be calculated.

Once a geometric series is confirmed to converge, calculating its sum is straightforward using a known formula. For a convergent geometric series, the sum \( S \) is given by:

\[ S = \frac{a}{1 - r} \]

where \( a \) is the first term, and \( r \) is the common ratio. In our exercise, the first term \( a \) is \( \frac{1}{7} \), and \( r \) is \( \frac{1}{7} \), both of which we substitute into the formula to find the sum.

Substituting the values, we get:

\[ S = \frac{\frac{1}{7}}{1 - \frac{1}{7}} = \frac{\frac{1}{7}}{\frac{6}{7}} = \frac{1}{6}. \]

This result indicates that the infinite series, though seemingly never-ending, sums up to a neat finite quantity. The formula for the sum is powerful because it allows us to compute the sum exactly without needing to calculate each individual term.